2021 NJC P1 Q5

Two straight corridors, \(P\) and \(Q\), each of width \(2\) m, meet at right angles. A banner is hung across the ceiling of the corridors using a taut string such that the string is parallel to the ground and always touches the inside corner of the wall at point \(B\). The string also touches the outer walls at variable points \(A\) and \(C\) respectively. In the position shown in the diagram, the acute angle between \(AC\) and the wall of corridor \(P\) is \(\frac{\pi }{4}+\theta \), where \(\theta \) is a sufficiently small angle.

1. Show that \(AC=2\left[ \frac{1}{\sin \left( \frac{\pi }{4}+\theta \right)}+\frac{1}{\cos \left( \frac{\pi }{4}+\theta \right)} \right]\).[2]
2. Hence show that
   \(AC\approx r+s{{\theta }^{2}}\)
   where \(r\) and \(s\) are constants to be determined.[5]